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2003
An empirical note on the random walk behaviour
and market efficiency of Latin American stock
markets
A. C. Worthington
University of Wollongong, [email protected]
H. Higgs
Griffith University
Publication Details
This paper was originally published as Worthington, AC and Higgs, H, An empirical note on the random walk behaviour and market
efficiency of Latin American stock markets, Empirical Economics Letters, 2(5), September 2003, 183-97. Journal home page here.
Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:
[email protected]
An empirical note on the random walk behaviour and market efficiency of
Latin American stock markets
Abstract
This note examines the weak-form market efficiency of Latin American equity markets. Daily returns for
Argentina, Brazil, Chile, Colombia, Mexico, Peru and Venezuela are examined for random walks using serial
correlation coefficient and runs tests, Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and
Kwiatkowski, Phillips, Schmidt and Shin (KPSS) unit root tests and multiple variance ratio (MVR) tests. The
results, which are in broad agreement across the approaches employed, indicate that none of the markets are
characterised by random walks and hence are not weak-form efficient, even under some less stringent random
walk criteria.
Keywords
Emerging markets, random walk hypothesis, market efficiency
Disciplines
Business | Social and Behavioral Sciences
Publication Details
This paper was originally published as Worthington, AC and Higgs, H, An empirical note on the random walk
behaviour and market efficiency of Latin American stock markets, Empirical Economics Letters, 2(5),
September 2003, 183-97. Journal home page here.
This journal article is available at Research Online: http://ro.uow.edu.au/commpapers/298
An empirical note on the random walk
behaviour and market efficiency of Latin
American stock markets
ANDREW C. WORTHINGTON*
School of Accounting and Finance, University of Wollongong, Wollongong, Australia
HELEN HIGGS
Department of Accounting, Finance amd Economics, Griffith University, Brisbane, Australia
This note examines the weak-form market efficiency of Latin American equity markets. Daily returns for
Argentina, Brazil, Chile, Colombia, Mexico, Peru and Venezuela are examined for random walks using serial
correlation coefficient and runs tests, Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski,
Phillips, Schmidt and Shin (KPSS) unit root tests and multiple variance ratio (MVR) tests. The results, which are
in broad agreement across the approaches employed, indicate that none of the markets are characterised by
random walks and hence are not weak-form efficient, even under some less stringent random walk criteria.
Keywords: Emerging markets, random walk hypothesis, market efficiency
JEL classifications: C12, C14, G14, G15.
INTRODUCTION
Much of the evidence regarding the random walk behaviour of stock returns has been
garnered from developed markets. While the focus of research has now shifted towards
emerging markets, largely in recognition of the valuable contribution efficient markets can
play in financial development and economic growth, stock markets in Latin America have
received less attention than that elsewhere. The evidence that does exist is incomplete in that
it focuses on a small number of markets, draws upon low frequency and short sample data,
and relies on a narrow range of empirical techniques. In evidence, Urrutia (1995), Ojah and
Karemera (1999) Karemera et al. (1999) examined random walk behaviour in only four Latin
American markets using just variance ratio tests, and while Haque’s et al. (2001) analysis
added another three markets, none of these studies employed data with a higher frequency
than weekly or with a sample longer than a decade. Barry and Rodriguez (1997), Grieb and
*
Correspondence to: Professor A.C. Worthington, School of Accounting and Finance, University of
Wollongong, Wollongong NSW 2522, Australia. Tel. +61 (0)2 4221 3616, Fax. +61 (0)2 4221 4297, email.
[email protected].
2
Worthington and Higgs
Reyes (1999), Pagán and Soydemir (2000, 2001) and Curci et al. (2002) have examined Latin
American stock markets from a similar perspective.
To meet this deficiency, this note examines the random walk behaviour of seven Latin
American stock markets (Argentina, Brazil, Chile, Columbia, Mexico, Peru and Venezuela)
using daily data for up to a fifteen-year period and three sets of alternative, though
complementary, testing procedures. The remainder of this note is divided into four sections.
The first section provides a description of the data employed in the analysis. The next section
discusses the empirical methodology used. The results are dealt with in the third section. The
paper ends with some concluding remarks in the final section.
DESCRIPTION AND PROPERTIES OF THE DATA
The data employed in the study is composed of market value-weighted equity indices for
seven emerging Latin American markets; namely, Argentina (ARG), Brazil (BRZ), Chile
(CHL), Columbia (COL), Mexico (MEX), Peru (PRU), Venezuela (VEN). All data is
obtained from Morgan Stanley Capital International (MSCI) and specified in US dollar terms.
The series encompass dissimilar sampling periods given the varying availability of each
index. The end date for all series is 28-May-2003 with ARG, BRZ, CHL and MEX
commencing on 31-Dec-1987 and COL, PRU and VEN on 31-Dec-1992. MSCI indices are
widely employed in the financial literature on the basis of the degree of comparability and
avoidance of dual listing, and are constructed to overcome problems associated with
infrequent or non-synchronous trading in markets.
Daily data is specified. The natural log of the relative price is computed for the daily intervals
to
produce
rt = log( pt pt
a
1
time
)×100 ,
series
of
continuously
compounded
returns,
such
that
where pt and pt-1 represent the stock index price at time t and t-1,
respectively. Table 1 presents a summary of descriptive statistics of the daily returns for the
seven markets. Sample means, maximums, minimums, standard deviations, skewness,
kurtosis and Jacque-Bera statistics and p-values are reported. The lowest mean returns are in
Columbia (-0.0001) Venezuela (0.0000) and the highest mean returns are for Argentina
(0.0004) and Mexico (0.0006). The lowest minimum returns are in Argentina (-0.9270) and
Venezuela (-0.7124) as are the highest maximum returns (0.4559 and 0.2137, respectively).
The standard deviations of returns range from 0.0127 (Chile) to 0.0401 (Argentina). On this
An empirical note on the random walk behaviour and market efficiency of Latin American stock markets
3
basis, of the seven markets the returns in Chile, Columbia and Peru are the least volatile, with
Venezuela, Brazil and Argentina being the most volatile.
<TABLE 1 HERE>
By and large, the distributional properties of all seven return series appear non-normal. Given
that the sampling distribution of skewness is normal with mean 0 and standard deviation of
6 T where T is the sample size, all of the return series, with the exception of Mexico and
Peru, are significantly skewed. Venezuela, Argentina, Chile and Brazil are negatively skewed,
indicating the greater probability of large deceases in returns than rises, while Columbia is
positively skewed, signifying the greater likelihood of large increases in returns than falls.
The kurtosis, or degree of excess, in all market returns is also large, ranging from 9.0184 for
Peru to 153.7496 for Venezuela, thereby indicating leptokurtic distributions. Given the
sampling distribution of kurtosis is normal with mean 0 and standard deviation of
24 T
where T is the sample size, then all estimates are once again statistically significant at any
conventional level. Finally, the calculated Jarque-Bera statistics and corresponding p-values
in Table 1 are used to test the null hypotheses that the daily distribution of market returns is
normally distributed. All p-values are smaller than the .01 level of significance suggesting the
null hypothesis can be rejected. None of these market returns are then well approximated by
the normal distribution.
EMPIRICAL METHODOLOGY
Random walk hypothesis
Consider the following random walk with drift process:
pt = p t 1 +
+
rt = pt =
+
(1)
t
or
(2)
t
where pt is the price of the index observed at time t,
change in the index and
t
is an arbitrary drift parameter, rt is the
is a random disturbance term satisfying E( t) = 0 and E(
t t-g)
= 0, g
0, for all t. Under the random walk hypothesis, a market is (weak-form) efficient if the most
recent price contains all available information and therefore the best predictor of future prices
is the most current price.
4
Worthington and Higgs
Within the random walk hypothesis, three successively more restrictive sub-hypotheses with
sequentially stronger tests for random walks exist (Campbell et al. 1997). The least restrictive
of these is that in a market that complies with a random walk it is not possible to use
information on past prices to predict future prices. That is, returns in a market conforming to
this standard of random walk are serially uncorrelated, corresponding to a random walk
hypothesis with dependent but uncorrelated increments. However, it may still be possible for
information on the variance of past prices to predict the future volatility of the market. A
market that conforms to these conditions implies that returns are serially uncorrelated,
corresponding with a random walk hypothesis with increments that are independent but not
identically distributed. Finally, if it is not possible to predict either future price movements or
volatility on the basis of information from past prices then such a market complies with the
most restrictive notion of a random walk. In this market, returns are serially uncorrelated and
conform to a random walk hypothesis with independent and identically distributed
increments.
This provides a number of complementary testing procedures for random walks or weak-form
market efficiency. To start with, the parametric serial correlation test of independence and the
non-parametric runs test can be used to test for serial independence in the series.
Alternatively, unit root tests can be used to determine if the series is difference or trend nonstationary as a necessary condition for a random walk. Finally, multiple variance ratio
procedures can focus attention on the uncorrelated residuals in the series, under assumptions
of both homoskedastic and heteroskedastic random walks.
Tests of serial independence
Two approaches are employed to test for serial independence in the returns. First, the serial
correlation coefficient test is a widely employed procedure that tests the relationship between
returns in the current period and those in the previous period. If no significant autocorrelations
are found then the series are assumed to follow a random walk. Second, the runs test
determines whether successive price changes are independent and unlike the serial correlation
test of independence, is non-parametric and does not require returns to be normally
distributed. Observing the number of ‘runs’ - or the sequence of successive price changes with
the same sign - in a sequence of price changes tests the null hypothesis of randomness. In the
approach selected, each return is classified according to its position with respect to the mean
return. That is, a positive change is when the return is greater than the mean, a negative
An empirical note on the random walk behaviour and market efficiency of Latin American stock markets
5
change when the return is less than the mean, and zero change when the return equals the
mean.
To perform this test A is assigned to each return that equals or exceeds the mean value and B
for the items that are below the mean. Let nA and nB be the sample sizes of items A and B
respectively. The test statistic is U, the total number of runs. For large sample sizes, that is
where both nA and nB are greater than twenty, the test statistic is approximately normally
distributed:
Z=
µU
U
(3)
U
where µU =
2nA nB
+ 1,
n
U
=
2nA nB (2n AnB n)
n 2 (n 1)
and n = n A + nB
Unit root tests
Three different unit root tests are used to test the null hypothesis of a unit root: namely, the
Augmented Dickey-Fuller (ADF) test, the Phillips-Peron (PP) test, and the Kwiatkowski,
Phillips, Schmidt and Shin (KPSS) test. To start with, the well-known ADF unit root test of
the null hypothesis of nonstationarity is conducted in the form of the following regression
equation:
pit =
0
+
1t +
0 p it 1 +
q
i
pit i +
(4)
it
i =1
where pit denotes the price for the i-th market at time t,
pit = pit
pit 1 ,
to be estimated, q is the number of lagged terms, t is the trend term,
coefficient for the trend,
0
is the constant, and
are coefficients
1
is the estimated
is white noise. MacKinnon’s critical values
are used in order to determine the significance of the test statistic associated with
0.
The PP
incorporates an alternative (nonparametric) method of controlling for serial correlation when
testing for a unit root by estimating the non-augmented Dickey-Fuller test equation and
modifying the test statistic so that its asymptotic distribution is unaffected by serial
correlation. Finally, the KPSS test differs from these other unit root tests in that the series is
assumed to be stationary under the null.
Multiple variance ratio tests
The multiple variance ratio (MVR) test as proposed by Chow and Denning (1993) is used to
detect autocorrelation and heteroskedasticity in the returns. Based on Lo and MacKinlay’s
6
Worthington and Higgs
(1988) earlier single variance ratio (VR) test, Chow and Denning (1993) adjusts the focus of
the tests from the individual variance ratio for a specific interval to one more consistent with
the random walk hypothesis by covering all possible intervals. As shown by Lo and
MacKinlay (1988), the variance ratio statistic is derived from the assumption of linear
relations in observation interval regarding the variance of increments. If a return series
follows a random walk process, the variance of a qth-differenced variable is q times as large
as the first-differenced variable. For a series partitioned into equally spaced intervals and
characterised by random walks, one qth of the variance of (pt - pt-q) is expected to be the same
as the variance of (pt – pt-1):
pt q ) = qVar ( p t
Var ( pt
(5)
pt 1 )
where q is any positive integer. The variance ratio is then denoted by:
1
Var ( pt pt q )
q
=
VR(q) =
Var ( pt pt 1 )
2
2
(q )
(1)
(6)
such that under the null hypothesis VR(q) = 1. For a sample size of nq + 1 observations (p0, p1,
…, pnq), Lo and Mackinlay’s (1988) unbiased estimates of
2
(1) and
2
(q) are
computationally denoted by:
nq
ˆ 2 (1) =
( pk
pk
1
k =1
µˆ ) 2
(7)
(nq 1)
and
nq
( pk
ˆ 2 (q) =
pk
q
qµˆ ) 2
k =q
where µˆ = sample mean of ( pt
h q(nq + 1 q)(1
(8)
h
q
)
nq
pt 1 ) and:
(9)
Lo and Mackinlay (1988) produce two test statistics, Z(q) and Z*(q), under the null hypothesis
of homoskedastic increments random walk and heteoskedastic increments random walk
respectively. If the null hypothesis is true, the associated test statistic has an asymptotic
standard normal distribution. With a sample size of nq + 1 observations (p0, p1, …,pnq) and
under the null hypothesis of homoskedastic increments random walk, the standard normal test
statistic Z(q) is:
An empirical note on the random walk behaviour and market efficiency of Latin American stock markets
Z (q) =
VRˆ (q ) 1
ˆ 0 (q)
7
(10)
where
2(2q 1)(q 1)
ˆ 0 (q) =
3q(nq)
1/ 2
(11)
The test statistic for a heteroskedastic increments random walk, Z*(q) is:
Z * (q) =
VRˆ (q ) 1
ˆ e (q)
(12)
where
ˆ e (q) = 4
q 1
k =1
1/ 2
2
k
1
q
ˆ
(13)
k
and
nq
(pj
ˆ =
k
j =( k +1)
pj
1
µˆ ) 2 ( p j
k
pj
µˆ ) 2
2
nq
j =1
k 1
(pj
pj
1
(14)
µˆ ) 2
Lo and MacKinlay’s (1988) procedure is devised to test individual variance ratios for a
specific aggregation interval, q, but the random walk hypothesis requires that VR(q) = 1 for all
q. Chow and Denning’s (1993) multiple variance ratio (MVR) test generates a procedure for
the multiple comparison of the set of variance ratio estimates with unity. For a single variance
ratio test, under the null hypothesis, VR(q) = 1, hence Mr(q) = VR(q) – 1 = 0. Consider a set of
m variance ratio tests {Mr(qi) i = 1,2,…,m}. Under the random walk null hypothesis, there
are multiple sub-hypotheses:
Hoi:
Mr(qi) = 0
for i = 1,2,…,m
H1i:
Mr(qi)
for any i = 1,2,…,m
0
(15)
The rejection of any one or more Hoi rejects the random walk null hypothesis. For a set of test
statistics, say Z(q), {Z(qi) i = 1,2,…,m}, the random walk null hypothesis is rejected if any
one of the estimated variance ratio is significantly different from one. Hence only the
maximum absolute value in the set of test statistics is considered. The core of the Chow and
Denning’s (1993) MVR test is based on the result:
PR{max ( Z (q1 ) ,..., Z (q m ) ) SMM ( ; m; T )} 1
(16)
8
Worthington and Higgs
where SMM( ;m;T) is the upper
point of the Standardized Maximum Modulus (SMM)
distribution with parameters m (number of variance ratios) and T (sample size) degrees of
freedom. Asymptotically when T approaches infinity:
lim SMM ( ; m; ) = Z
T!
where Z
*
/2
*
(17)
/2
= standard normal distribution and
*
= 1 (1
)1/ m . Chow and Denning (1993)
control the size of the MVR test by comparing the calculated values of the standardized test
statistics, either Z(q) or Z*(q) with the SMM critical values. If the maximum absolute value of,
say Z(q) is greater than the SMM critical value than the random walk hypothesis is rejected.
Importantly, the rejection of the random walk under homoskedasticity could result from either
heteroskedasticity and/or autocorrelation in the equity price series. If the heteroskedastic
random walk is rejected than there is evidence of autocorrelation in the equity series. With the
presence of autocorrelation in the price series, the first order autocorrelation coefficient can be
estimated using the result that Mˆ r (q ) is asymptotically equal to a weighted sum of
autocorrelation coefficient estimates with weights declining arithmetically:
q 1
k
ˆ (k )
q
(18)
Mˆ r (2) VRˆ (2) 1 = ˆ (1)
(19)
Mˆ r (q) = 2
1
k =1
where q = 2 :
EMPIRICAL RESULTS
Table 2 provides two sets of test statistics. The first set includes the statistics and p-values for
the tests of serial independence, namely, the parametric serial correlation coefficient and the
nonparametric one sample runs test. The null hypothesis in the former is for no serial
correlation while in the latter it is the random distribution of returns. The second set of tests is
unit root tests and comprises the ADF and PP t-statistics and p-values and the KPSS LMstatistic and asymptotic significance. In the case of the former the null hypothesis of a unit
root is tested against the alternative of no unit root (stationary). For the latter, the null
hypothesis of no unit root is tested against the alternative of a unit root (non-stationary).
<TABLE 2 HERE>
Turning first to the tests of independence, the null hypotheses of no serial correlation for
Brazil, Chile, Columbia, Mexico, Peru and Venezuela are rejected at the .01 level or higher,
An empirical note on the random walk behaviour and market efficiency of Latin American stock markets
9
while that for Argentina is rejected at the .05 level or higher. The significance of the
autocorrelation coefficient indicates that the null hypothesis of weak-form market efficiency
may be rejected and we may infer that all seven Latin American markets are weak-form
inefficient.
With the exception of Argentina, all of the coefficients are positive indicating persistence in
returns, with persistence being higher in Columbia (0.3390) and Chile (0.2270) and lower in
Brazil (0.1520) and Mexico (0.1230). The average persistence is 0.1858 across all six
markets. For Argentina the serial correlation coefficient of -0.0310 is indicative of a mean
reversion process. However, it should be noted that over shorter horizons the six markets
exhibiting persistence (and Argentina exhibiting mean-reversion) could also display meanreversion (persistence). In terms of the runs tests, the negative z-values for all of the markets,
including Argentina, indicates that the actual number of runs falls short of the expected
number of runs under the null hypothesis of return independence at the .01 level or lower for
all markets. These indicate positive serial correlation. We likewise reject the null hypothesis
of weak-form efficiency when employing the nonparametric assumptions entailed in runs
tests. By way of comparison, Karemera et al. (1999) also used runs tests (though with
monthly returns) to conclude that Argentina, Brazil and Mexico were weak form efficient
from an international investor’s perspective (when measured in US dollars) while Brazil and
Mexico were weak-form efficient in local currency terms. Urrutia’s (1995) runs tests likewise
failed to reject the null hypothesis of independence for Argentina, Brazil, Chile and Mexico.
The unit root tests in Table 2 are also supportive of the hypothesis that Latin American equity
markets are not weak form efficient. The ADF and PP t-statistics reject the null hypotheses of
a unit root at the .01 level or lower, thereby indicating that all of the return series examined
are stationary. For the KPSS tests of the null hypothesis of no unit root, the LM-statistic
exceeds the asymptotic critical value at the .05 level for Chile (0.6839) and at the .10 level for
Argentina (0.3925) and Mexico (0.5410). As a necessary condition for a random walk, the
ADF and PP unit root tests reject the requisite null hypothesis in the case of all seven markets,
while the KPSS unit root tests fail to reject the required null with the exception of the
Argentina, Chile and Mexico.
Table 3 presents the results of the multiple variance ratio tests of returns in the seven Latin
American markets. The sampling intervals for all markets are 2, 5, 10 and 20 days,
corresponding to one-day, one week, one fortnight and one month calendar periods. For each
interval Table 3 presents the estimates of the variance ratio VR(q) and the test statistics for the
10
Worthington and Higgs
null hypotheses of homoskedastic, Z(q) and heteroskedastic, Z*(q) increments random walk.
Under the multiple variance ratio procedure, only the maximum absolute values of the test
statistics are examined. For sample sizes exceeding at least 2,714 observations (Columbia,
Peru and Venezuela) and where m = 4, the critical value for these test statistics is 2.49 at the
.05 level of significance. For each set of multiple variance ratio tests, an asterisk denotes the
maximum absolute value of the test statistic that exceeds this critical value and thereby
indicates whether the null hypothesis of a random walk is rejected.
<TABLE 3 HERE>
Consider the results for Mexico. The null hypothesis that daily equity returns follow a
homoskedastic random walk is rejected at Z(2) = 7.8833. Rejection of the null hypothesis of a
random walk under homoskedasticity for a 2-day period is also a test of the null hypothesis of
a homoskedastic random walk under the alternative sampling periods and we may therefore
conclude that Mexican equity returns do not follow a random walk. However, rejection of the
null hypothesis under homoskedasticity could result from heteroskedasticity and/or
autocorrelation in the return series. After a heteroskedastic-consistent statistic is calculated,
the null hypothesis is also rejected at Z*(2) = 3.6223. The heteroskedastic random walk
hypothesis is thus rejected because of autocorrelation in the daily increments of the returns on
Mexican equity. We may conclude that the Mexican equity market is not weak form efficient.
Further, Lo and MacKinlay (1988) show that for q=2, estimates of the variance ratio minus
one and the first-order autocorrelation coefficient estimator of daily price changes are
asymptotically equal [Mexico’s serial correlation coefficient in Table 2 is 0.1230]. On this
basis, the estimated first order autocorrelation coefficient is 0.1244 corresponding to the
estimated variance ratio VR̂(2) of 1.1244 (i.e. 1.1244 - 1.0000). Further, where VRˆ (2) < 1 a
mean reverting process is indicated, whereas when VRˆ (2) > 1 persistence is suggested. This
indicates there is positive autocorrelation (or persistence) in Mexican equity returns over the
long horizon.
By way of comparison, observe the results for Argentina. The null hypothesis that daily
equity returns follow a homoskedastic random walk is rejected at Z(5) = -4.8571 which in
absolute terms is greater than the critical value of 2.49. This likewise suggests that the
Argentinean equity market is weak form inefficient. However, the null hypothesis of a
heteroskedastic random walk is not rejected [Z*(q)=-1.5782]. This indicates that rejection of
the null hypothesis of a homoskedastic random walk could be the result, at least in part, of
An empirical note on the random walk behaviour and market efficiency of Latin American stock markets
11
heteroskedasticity in the returns, and cannot be assigned exclusively to autocorrelation in
returns. In addition, since VRˆ (2) < 1 for Argentina we can infer that its market returns are
characterised by mean reversion (i.e. 0.9698 - 1.0000 = 0.0302).
Of the seven markets, the multiple variance ratios testing procedure rejects the null hypothesis
of a random walk under assumptions of both homoskedasticity and heteroskedasticity for all
except Argentina. We may then conclude that none of these markets are weak form efficient.
With Argentina the null hypothesis of a homoskedastic random walk is rejected, but not that
for a heteroskedastic random walk. This infers that the observed random walk violation could
be the result of heteroskedasticity and autocorrelation in daily returns, thereby corresponding
to a less stringent version of the random walk hypothesis. Nevertheless, the multiple variance
ratio technique indicates the presence of positive autocorrelation (or persistence) in six
markets and negative autocorrelation (or mean reversion) for Argentina and thereby provides
comparable evidence to the results of the serial correlation coefficients and runs tests. They do
however strongly contradict the earlier evidence provided by Urrutia (1995) Ojah and
Karemera (1999) that Argentina, Brazil, and Chile were weak form efficient, and by Urrutia
(1995) and Karemera et al. (1999) that Mexico was also weak form inefficient. They do,
however, substantiate Haque et al. (2001) conclusion that all of these markets are no weak
form efficient on the basis of testing the earlier Lo and MacKinlay (1988) single variance
ratio procedure using weekly returns.
CONCLUSION
This note examines the weak form market efficiency of seven Latin American equity markets;
namely, Argentina, Brazil, Chile, Columbia, Mexico, Peru and Venezuela. Three different
procedures are employed to test strict and less strict versions of the random walk hypothesis
in daily returns: (i) the parametric serial correlation coefficient and the nonparametric runs
test are used to test for serial correlation; (ii) Augmented Dickey-Fuller, Phillips-Perron and
Kwiatkowski, Phillips, Schmidt and Shin unit root tests are used to test for non-stationarily as
a necessary condition for a random walk; and (iii) multiple variance test statistics are used to
test for random walks under varying distributional assumptions. The results for the tests of
serial correlation are in broad agreement, conclusively rejecting the presence of random walks
in daily returns in the seven emerging markets. Similarly, the unit root tests conclude that unit
roots, as necessary conditions for a random walk, are absent from all of the return series.
Finally, the multiple variance ratio procedure conclusively rejects the presence of random
12
Worthington and Higgs
walks in any Latin American market, though a random walk in the Argentinean market is
rejected under less restrictive criteria than the remaining markets.
REFERENCES
Barry, C. and Rodriguez, M. (1997) Risk, return and performance of Latin America’s equity
markets, 1975-1995, Latin American Business Review, 1(1), 51-76.
Campbell, J.Y. Lo, A.W. and MacKinlay, A.C. (1997) The Econometrics of Financial
Markets, Princeton, Princeton University Press.
Chow. K.V. and Denning, K. (1993) A simple multiple variance ratio test, Journal of
Econometrics, 58(3), 385-401.
Curci, R. Grieb, T. and Reyes, M.G. (2002) Mean and volatility transmission for Latin
American equity markets, Studies in Economics and Finance, 20(2), 39-57.
Grieb, T. and Reyes, M.G. (1999) Random walk tests for Latin American equity indexes and
individual firms, Journal of Financial Research, 22(4), 371-383.
Haque, M. Hassan, M.K. and Varela, O. (2001) Stability, volataility, risk premiums and
predictability in Latin American emerging stock markets, Quarterly Journal of Business
and Economics, 40(3-4), 23-44.
Karemera, D. Ojah, K. and Cole, J.A. (1999) Random walks and market efficiency tests:
Evidence from emerging equity markets, Review of Quantitative Finance and Accounting,
13(2), 171-188.
Lo, A. and MacKinlay, A.C. (1988) Stock market prices do not follow random walks:
Evidence from a simple specification test, Review of Financial Studies, 1(1), 41-66.
Ojah, K. amd Karemera, D. (1999) Random walks and market efficiency tests of Latin
American Emerging equity markets: A revisit, The Financial Review, 34(1), 57-72.
Pagán, J.A. and Soydemir, G.A. (2001) Response asymmetries in the Latin American equity
markets, International Review of Financial Analysis, 10(1), 175-185.
Pagán, J.A. and Soydemir, G.A. (2000) On the linkages between equity markets in Latin
America, Applied Economics Letters, 7(3), 207-210.
Urrutia, J.L. (1995) Tests of random walk and market efficiency for Latin American emerging
markets, Journal of Financial Research, 18(3), 299-309.
TABLE 1. Descriptive statistics for Latin American markets
JarqueJB p-value
Bera
ARG
4019
4.52E-04
0.4559 -0.9270
0.0401 -2.8730 95.1709 1.43E+06
0.0000
31-Dec-1987 28-May-2003
BRZ
4019
3.98E-04
0.2123 -0.2635
0.0288 -0.4078 10.6391 9.88E+03
0.0000
31-Dec-1987 28-May-2003
CHL
4019
4.19E-04
0.0870 -0.1623
0.0127 -0.4897 14.1018 2.08E+04
0.0000
31-Dec-1987 28-May-2003
COL
2714 -8.87E-05
0.1329 -0.0735
0.0132
0.3010 11.0072 7.29E+03
0.0000
31-Dec-1992 28-May-2003
MEX 31-Dec-1987 28-May-2003
4019
6.87E-04
0.1784 -0.2176
0.0196 -0.0669 15.4183 2.58E+04
0.0000
PRU
2714
2.69E-04
0.1065 -0.0930
0.0160
0.0579
9.0184 4.10E+03
0.0000
31-Dec-1992 28-May-2003
VEN
2714
8.45E-06
0.2137 -0.7124
0.0284 -5.3078 153.7496 2.58E+06
0.0000
31-Dec-1992 28-May-2003
Notes: ARG – Argentina, BRZ – Brazil, CHL – Chile, COL – Columbia, MEX – Mexico, PRU – Peru, VEN – Venezuela. JB – JarqueBera. Critical values for significance of skewness and kurtosis at the .05 level are 0.0757 and 0.1514 for ARG, BRZ, CHL and MEX and
0.0921 and 0.1843 for COL, PRU and VEN.
Market
Start
End
Observations
Mean
Maximum Minimum Std. Dev. Skewness Kurtosis
TABLE 2. Tests of independence and unit root tests for Latin American markets
Serial correlation
Runs test
Cases <
mean
Cases
mean
Total
cases
Unit root tests
Number
of runs
Runs Zvalue
ADF
t-statistic
PP
KPSS LM- KPSS
p-value
statistic significance
ARG
-0.0310
0.0247 4.52E-04
2106
1913
4019
1867
-4.3916
0.0000
-38.1018
0.0000 -66.0677
0.0001
0.3925 0.1000
BRZ
–
0.1520
0.0000 3.98E-04
2054
1965
4019
1791
-6.8979
0.0000
-54.3501
0.0001 -55.0195
0.0001
0.1234
CHL
0.2270
0.0000 4.19E-04
2126
1893
4019
1585
-13.2568
0.0000
-50.1775
0.0001 -50.2673
0.0001
0.6839 0.0500
COL
–
0.3390
0.0000 -8.87E-05
1315
1399
2714
1043
-12.0569
0.0000
-36.5759
0.0000 -37.4777
0.0000
0.1244
MEX
0.1230
0.0000 6.87E-04
2074
1945
4019
1775
-7.3727
0.0000
-55.9574
0.0001 -55.8887
0.0001
0.5410 0.0500
PRU
–
0.1810
0.0000 2.69E-04
1404
1310
2714
1245
-4.2816
0.0000
-43.3626
0.0000 -43.2109
0.0000
0.1628
VEN
–
0.0930
0.0000 8.45E-06
1454
1260
2714
1185
-6.4093
0.0000
-45.5321
0.0001 -45.3040
0.0001
0.0414
Notes: ARG – Argentina, BRZ – Brazil, CHL – Chile, COL – Columbia, MEX – Mexico, PRU – Peru, VEN – Venezuela. For Augmented Dickey-Fuller (ADF) tests hypotheses
are H0: unit root, H1: no unit root (stationary). The lag orders in the ADF equations are determined by the significance of the coefficient for the lagged terms. Intercepts only in the
series. The Phillips-Peron (PP) unit root test hypotheses are H0: unit root, H1: no unit root (stationary). Intercepts only in the series. The Kwiatkowski, Phillips, Schmidt and Shin
(KPSS) unit root test hypotheses are H0: no unit root (stationary), H1: unit root. The asymptotic critical values for the KPSS LM test statistic at the .10, .05 and .01 levels are
0.3470, 0.4630 and 0.7390 respectively.
Market Coefficient p-value
Mean
p-value
ADF
p-value
PP
t-statistic
TABLE 3. Multiple variance ratio tests for Latin American markets
Market Statistics q = 2
q=5
q = 10
q = 20
ARG VRq
0.9698
0.8321
0.7445
0.7630
Zq
-1.9147 *-4.8571 -4.7980 -3.0226
Z*q
-0.6659 -1.4829 -1.5782 -1.0872
BRZ VRq
1.1530
1.3386
1.4444
1.5376
Zq
9.6965 *9.7974
8.3445
6.8571
Z*q
5.0969 *5.3539
4.9252
4.3488
CHL VRq
1.2299
1.3986
1.5281
1.7725
Zq
*14.5756 11.5350
9.9155
9.8541
Z*q
*9.4875
7.7593
7.0012
7.4213
COL VRq
1.3402
1.8039
2.0702
2.4869
Zq
17.7205 *19.1145 16.5124 15.5860
Z*q
10.1606 *11.9481 11.0265 11.0981
MEX VRq
1.1244
1.1939
1.2105
1.3425
Zq
*7.8833
5.6113
3.9516
4.3690
Z*q
*3.6223
2.8368
2.1745
2.5795
PRU VRq
1.1818
1.2814
1.2651
1.3679
Zq
*9.4709
6.6907
4.0902
3.8568
Z*q
*5.8074
4.3022
2.7807
2.7895
VEN VRq
1.1342
1.1300
1.1495
1.1610
Zq
*6.9890
3.0900
2.3061
1.6876
Z*q
*4.0589
1.8932
1.5833
1.2960
Notes: ARG – Argentina, BRZ – Brazil, CHL – Chile, COL –
Columbia, MEX – Mexico, PRU – Peru, VEN – Venezuela.
VR(q) – variance ratio estimate, Z(q) - test statistic for null
hypothesis of homoskedastic increments random walk, Z* (q)
- test statistic for null hypothesis of heteroskedastic
increments random walk; the critical value for Z(q) and Z*(q)
at the 5 percent level of significance is 2.49, asterisk indicates
significance at this level; Sampling intervals (q) are in days.